Some Analytical Approximate Solutions Of Burgers Equation

Burgers equation,as a kind of nonlinear partial differential equation,is mainly used to simulate the propagation and reflection of shock wave.It has important applications in various fields of Applied Mathematics,especially in hydrodynamics,nonlinear acoustics and aerodynamics.In general,it is difficult to find the exact solution by nonlinear partial differential method.Therefore,it is of theoretical and practical significance to study the analytical approximate solution of nonlinear partial differential equations.This paper is based on the principle of homogeneous balance,the exponential function method,the hyperbolic function expansion method and the Riccati equation rational expansion method are used to solve the one-dimensional Burgers equation respectively.With the help of the mathematical software Maple and Mathematica,various analytical approximate solutions of traveling wave solution,solitary wave solution,kink solitary wave solution and rational form solution are obtained;The same method is applied to Burgers equation in two-dimensional case,and the analytical approximate solution is obtained;In order to observe the structure of the solution more intuitively,the three-dimensional image of the equation is obtained by taking some special values.These solutions and images will be of positive significance to the study of Burgers equation.Finally,the difference scheme is used to simulate two kinds of analytical solutions in one-dimensional case,and the results are compared with the analytical approximate solutions.The validity of the three analytical approximate methods proposed in this paper is verified,which can be used to solve the practical problems described by Burgers equation.